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18th GAMM-Seminar Leipzig on
Multigrid and related methods for optimization problems

Max-Planck-Institute for Mathematics in the Sciences
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  18th GAMM-Seminar
January, 24th-26th, 2002
 
     
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  Abstract F. Schieweck, Sat, 10.00-10.25 Previous Contents Next  
  A new Stokes solver based on Lagrange multipliers
F. Schieweck (Otto-von-Guericke Universität Magdeburg)

We consider the discrete Stokes problem generated by a finite element method with continuous Q_{r}-elements for the velocity components and discontinuous P_{r-1}-elements for the pressure. Our aim is to construct an efficient solver for the arising indefinite system of equations. The idea is to solve the problem for a larger velocity space, which contains also discontinuous velocity modes, and to handle the desired continuity of the velocity by means of Lagrange multipliers. This allows a decoupling into element-by-element calculations. Another key ingredient of our method is the orthogonal decomposition of the larger velocity space into a discrete divergence-free subspace of the original conforming velocity space and a discontinuous complement space. Thus, the solution of the Stokes problem splits into two symmetric positive definite (s.p.d.) subproblems. The first one is an s.p.d. problem for the discrete divergence-free part of the velocity where the pressure is eliminated. This problem can be solved efficiently by a multigrid method. The second subproblem is an s.p.d. problem for the Lagrange multipliers. It can be shown that the condition number of the corresponding operator is O(1) for h -> 0. Therefore, a CG-method is an efficient solver. The ideas can be applied also to the Navier-Stokes equations.
 

 
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Concept, Design and Realisation
[O->]Jens Burmeister (Uni Kiel), Kai Helms (MPI Leipzig)
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